半导体电子结构

20 Electronic Structure of Semiconductors Since the invention of the transistor in 1947–1948, and especially the start of the mass production of integrated circuits and microprocessors, semiconductor devices have played an ever increasing role in modern information technology as well as many other applied fields. The optical properties of semiconductors, which set them distinctly apart of metals, are also exploited in a wide range of applications. In addition to being very important for materials science, the study of semiconductors is of great interest for fundamental research as well, as a great number of new phenomena can be observed in them. For example, the discovery of the quantum Hall effect arose from the possibility of creating semiconductor heterojunctions in which the electron gas is practically confined to a two-dimensional region next to the interface. This opened the way to the study of the properties rooted in the two-dimensional character of the system. Besides, very high purity materials can be fabricated from semiconductors, which is a prerequisite to studying certain physical phenomena. To derive the physical properties of semiconductors from first principles, we have to familiarize ourselves with the characteristic properties of their elec- tronic structure. The methods of band-structure calculation presented in the previous chapter can be equally applied to metals and insulators, so they can just as well be used for the description of the electronic structure of semi- conductors. However, the devices designed to exploit the physical properties of semiconductors practically never use pure crystals: the number of charge carriers is controlled by doping the semiconductor components with impuri- ties. Therefore we shall also study the states formed around impurities, and how the energy spectrum is modified by them. The phenomena required to understand the operation of semiconductor devices, as well as the particular conditions arising close to the interfaces and in inhomogeneous semiconductor structures will be presented in Chapter 27, after the discussion of transport phenomena. 196 20 Electronic Structure of Semiconductors 20.1 Semiconductor Materials As mentioned earlier, a characteristic property of semiconductors is that their resistivity falls between that of metals and insulators. An even more interesting feature is the temperature dependence of resistivity. In contrast to metals and semimetals, it increases exponentially with decreasing temperature in pure semiconductors: � ∝ exp(ε0/kBT ). The Hall coefficient RH is positive in several cases, which can be interpreted by assuming that the principal charge carriers in these materials are not electrons but holes. It is also known from the behavior of the Hall coefficient that, in contrast to metals, the number of carriers depends strongly on temperature. This is in perfect agreement with a band structure in which the bands that are filled completely in the ground state are separated from the completely empty bands by a forbidden region, a gap, whose width εg is finite but not much larger than the thermal energy kBT . In semiconductors the highest band that is completely filled in the ground state is called the valence band, while the lowest completely empty band is called the conduction band. Current can flow only when charge carriers – electrons in the conduction band and holes in the valence band – are generated by thermal excitation. As we shall see, the probability of generating carriers by thermal excita- tion is proportional to exp(−εg/2kBT ) because of the finite gap. If the band gap is around εg ∼ 5 eV, this probability is about e−100 or 10−45 at room temperature (kBT ∼ 0.025 eV). Since the total electron density is on the or- der of 1023 per cm3, the conduction band contains practically no electrons. However, when the gap is only 1 eV, the excitation probability is 10−9 because of the exponential dependence, and so the density of thermally excited elec- trons is 1014/cm3. Such a number of mobile charges gives rise to observable phenomena. The resistivity of semiconductors is very sensitive to the presence of impu- rities. For this reason, pure stoichiometric semiconductors are of much smaller practical importance than doped ones. Nonetheless we shall start our discus- sion with pure materials. 20.1.1 Elemental Semiconductors As mentioned in Chapter 17, elemental semiconductors are located in the even-numbered groups of the periodic table, to the right of transition met- als. Certain elements of the group IIIA (boron group)1 and the group VA (nitrogen group) also occur in compound semiconductors. Table 20.1 shows the electronic configuration of the outermost shell in the atomic state of the elements of interest. Elemental semiconductors are found in group IVA (carbon group), and group VIA (oxygen group, chalcogens). The first element of the carbon group, 1 Following the conventions of semiconductor physics, we shall often use only the traditional designation for the groups of the periodic table in this chapter. 20.1 Semiconductor Materials 197 Table 20.1. Elements occurring in semiconductor materials, and the configuration of their outermost shell in the atomic state Group IIB Group IIIA Group IVA Group VA Group VIA B 2s2 2p1 C 2s2 2p2 N 2s2 2p3 O 2s2 2p4 Al 3s2 3p1 Si 3s2 3p2 P 3s2 3p3 S 3s2 3p4 Zn 3d10 4s2 Ga 4s2 4p1 Ge 4s2 4p2 As 4s2 4p3 Se 4s2 4p4 Cd 4d10 5s2 In 5s2 5p1 Sn 5s2 5p2 Sb 5s2 5p3 Te 5s2 5p4 Hg 5d10 6s2 Tl 6s2 6p1 Pb 6s2 6p2 Bi 6s2 6p3 Po 6s2 6p4 carbon, has several allotropes. Even though not a semiconductor, diamond is of great interest here as it features the same type of bonding and structure as the other, semiconducting, elements of the group. The sp3 hybrid states given in (4.4.52) of the outermost s- and p-electrons form four covalent bonds in the directions of the vertices of a regular tetrahedron. This leads to the diamond lattice shown in Fig. 7.16. As discussed in Chapter 4, in covalently bonded materials the density of electrons is highest in the region between the two atoms. This was illustrated for germanium in Fig. 4.5, where a section of the spatial distribution of the valence electrons was shown in the vicinity of the line joining neighboring atoms. Of the elements of group IVA, the band structure of diamond and gray tin (α-Sn) – both determined by the LCAO method – were shown in Fig. 17.10. The band structure of silicon and germanium will be discussed in detail and illustrated later (Figs. 20.2 and 20.5). In each case, there are further narrow and completely filled bands below the shown bands. The lowest four of the shown bands – which are partially degenerate along certain high-symmetry directions – are formed by the s- and p-electrons that participate in cova- lent bonding. In the ground state these bands are completely filled, since the primitive cell contains two electrons with four valence electrons each. Except for α-Sn, these four bands are separated by a finite gap from the higher-lying ones (that are completely empty in the ground state). Pure diamond is an insulator as its energy gap is 5.48 eV. However, when doped, it exhibits the characteristic properties of semiconductors. The two next elements of the carbon group, silicon (Si) and germanium (Ge) are good semiconductors even in their pure form; their energy gap is close to 1 eV. The measured data show a slight nonetheless clear temperature dependence. This is due to the variations of the lattice constant, which modify the overlap between the electron clouds of neighboring atoms, leading to an inevitable shift of the band energies. The energy gap measured at room temperature and the value extrapolated to T = 0 from low-temperature measurements are listed in Table 20.2. The fourth element of this group is tin. Among its several allotropes gray tin is also a semiconductor, but it has hardly any practical importance. The 198 20 Electronic Structure of Semiconductors Table 20.2. Energy gap at room temperature and at low temperatures for group IVA elements Element εg(300K) (eV) εg(T = 0) (eV) C 5.48 5.4 Si 1.110 1.170 Ge 0.664 0.744 energy gap is not listed in the table because there is no real gap in the band structure. Nonetheless, as mentioned in Chapter 17, it behaves practically as a semiconductor, since the density of states is very low at the bottom of the conduction band, therefore electrons excited thermally at room temperature do not occupy these levels but a local minimum of the conduction band. Although this minimum is located 0.1 eV higher, it has a larger density of states. It is readily seen from the band structure and the values given in the table that the gap decreases toward the high end of the column. Tin is followed by lead, a metal whose band structure was shown in Fig. 19.9. Among the elements of the oxygen group, selenium (Se) and tellurium (Te) are the only semiconductors. The gap is 1.8 eV for Se and 0.33 eV for Te. In these materials the outermost s- and p-electrons can form two covalent bonds located along a helix, as shown in Fig. 7.24(a). The relatively weak van der Waals forces between the chains are strong enough to ensure that selenium and tellurium behave as three-dimensional materials. 20.1.2 Compound Semiconductors Among covalently bonded compounds quite a few are semiconductors. For that each s- and p-electron form saturated covalent bonds, the compound has to be built up of cations and anions that give, on the average, four electrons to the tetrahedrally coordinated bonds. This can be ensured in several different ways. The simplest possibility is to build a compound of two elements of the carbon group (group IVA) – or else an element of the boron group (group IIIA) can be combined with one of the nitrogen group (group VA), or an element of the zinc group (group IIB) with one of the oxygen group (group VIA). Among the compounds of the elements of the carbon group, silicon carbide (SiC, also called carborundum) is particularly noteworthy. Its energy gap is 2.42 eV. In stoichiometric composition it is a good insulator, but a small excess of carbon or other dopants turn it into a good semiconductor. For applications, III–V (AIIIBV) and II–VI (AIIBVI) semiconductors are more important. As their names show, these are compounds of elements in groups IIIA and VA, 20.1 Semiconductor Materials 199 and IIB and VIA. Table 20.3 shows the room-temperature energy gap for a small selection of them. Table 20.3. Energy gap for III–V, II–VI, and I–VII semiconductors at room tem- perature III–V compound εg (eV) II–VI compound εg (eV) I–VII compound εg (eV) AlSb 1.63 ZnO 3.20 AgF 2.8 GaP 2.27 ZnS 3.56 AgCl 3.25 GaAs 1.43 ZnSe 2.67 AgBr 2.68 GaSb 0.71 CdS 2.50 AgI 3.02 InP 1.26 CdSe 1.75 CuCl 3.39 InAs 0.36 CdTe 1.43 CuBr 3.07 InSb 0.18 HgS 2.27 CuI 3.11 A large number of III–V and II–VI semiconductors crystallize in the spha- lerite structure. The prototype of this structure, sphalerite (zinc sulfide) is a semiconductor itself. As shown in Fig. 7.16, this structure can be derived from the diamond structure by placing one kind of atom at the vertices and face centers of a fcc lattice, and the other kind of atom at the center of the four small cubes. The structure can also be considered to be made up of two interpenetrating fcc sublattices displaced by a quarter of the space diagonal. Thus each atom is surrounded tetrahedrally by four of the other kind. Here, too, s- and p-electrons participate in bonding, however the bond is not purely covalent on account of the different electronegativities of the two atoms. In AIIIBV semiconductors the ion cores A3+ and B5+ stripped of their s- and p-electrons do not attract electrons in the same way. Unlike in germanium (Fig. 4.5), the wavefunction of the electrons forming the covalent bond – and hence the density of the electrons – is not symmetric with respect to the position of the two ions, as shown in Fig. 4.7 for GaP: it is biassed toward the B5+ ions. This gives a slight ionic character to the bond. The asymmetry is even more pronounced for II–VI semiconductors, and the density maximum of the binding electrons is now even closer to the element of group VIA, as illustrated in Fig. 4.7 for ZnSe. The bond is thus more strongly ionic in character. The elements of the carbon group, for which no such asymmetry occurs, are called nonpolar semiconductors, while III–V and II–VI compounds are polar semiconductors. Even more strongly polar are the compounds of the elements of groups IB and VIIA. Some halides of noble metals behave as semiconductors, even though their gap is rather large. Copper compounds feature tetrahedrally coordinated bonds, but silver compounds do not: they crystallize in the sodium chloride structure. 200 20 Electronic Structure of Semiconductors It is worth comparing the properties of these compounds with those of alkali halides (formed by elements of groups IA and VIIA). In the latter the difference between the electronegativities of the two constituents is so large that purely ionic bonds are formed instead of polarized covalent bonds. Con- sequently the structure is also different, NaCl- or CsCl-type, and their gap (listed in Table 20.4 for a few alkali halides) is much larger than in any pre- viously mentioned case. Table 20.4. Energy gap in some alkali halides Compound εg (eV) Compound εg (eV) Compound εg (eV) LiF 13.7 LiCl 9.4 LiBr 7.6 NaF 11.5 NaCl 8.7 NaBr 7.5 KF 10.8 KCl 8.4 KBr 7.4 RbF 10.3 RbCl 8.2 RbBr 7.4 CsF 9.9 CsCl 8.3 CsBr 7.3 In addition to those listed above, there exist further compound semi- conductors with non-tetrahedrally coordinated covalent bonds. Tin and lead (group IVA) form semiconducting materials with elements of the oxygen group (group VIA) that crystallize in the NaCl structure. Elements of group IVA may also form compound semiconductors with alkaline-earth metals (group IIA) in the composition II2–IV (AII2 BIV). Semiconductor compounds with a non-1:1 composition can also be formed by IB and VIA, or IIB and VA elements. The best known examples are Cu2O (εg = 2.17 eV) and TiO2 (εg = 3.03 eV). The gap is much narrower in some other compounds, as listed in Table 20.5. Table 20.5. Energy gap in some compound semiconductors at low temperatures Compound εg (eV) Compound εg (eV) SnS 1.09 Mg2Si 0.77 SnSe 0.95 Mg2Ge 0.74 SnTe 0.36 Mg2Sn 0.36 PbO 2.07 Cu2O 2.17 PbS 0.29 Ag2S 0.85 PbSe 0.14 Zn3As2 0.86 PbTe 0.19 Cd3P2 0.50 Semiconducting properties are also observed in various nonstoichiometric compounds and even amorphous materials. 20.2 Band Structure of Pure Semiconductors 201 20.2 Band Structure of Pure Semiconductors According to the foregoing, in the ground state of semiconductors the com- pletely filled valence band is separated from the completely empty conduction band by a narrow gap. We shall first examine the band structure of the two best known semiconductors, silicon and germanium, focusing on how the en- ergy gap is formed and how the bands closest to the chemical potential can be characterized. To understand the band structure, we shall follow the steps outlined in Chapter 18: start with the empty-lattice approximation, and deter- mine how degeneracy is lifted and how gaps appear in the nearly-free-electron approximation. We shall then present the theoretical results based on more accurate calculations, and the experimental results. 20.2.1 Electronic Structure in the Diamond Lattice We have seen that the elemental semiconductors of the carbon group crystal- lize in a diamond structure. Since the Bravais lattice is face-centered cubic, the Brillouin zone is the truncated octahedron depicted in Fig. 7.11. Below we shall repeatedly make reference to the special points and lines of the Brillouin zone, therefore we shall recall them in Fig. 20.1(a). kx � � X � � �' L W Q K U ky kz � � � 0 1 2 3 4 L � � X �U K, 000 000000 2¯00 1¯ 1¯ 1¯ 8 (111) 6 (200) 02¯ 2¯ 002¯ 02¯0{} 1¯ 1¯ 1¯ 1¯ 1¯ 1¯ 1¯ 1¯ 1¯ { 1¯ 1¯ 1¯1¯ 1¯ 1¯1¯ 1¯ 1¯ 1¯ 1¯ 1¯ {1¯ 1¯ 1¯1¯ 1¯ 1¯1¯ 1¯ 1¯ 1¯ 1¯ 1¯ 1¯ 1¯ 1¯ 1¯ 1¯ 1¯ {1¯ 1¯ 1¯1¯ 1¯ 1¯1¯ 1¯ 1¯ 1¯ 1¯ 1¯ { }002¯ 02¯0 2¯00 ( )a ( )b Fig. 20.1. (a) Brillouin zone of the face-centered cubic lattice of the diamond structure with the special points and lines. (b) Band structure in the empty-lattice approximation, with the energy given in units of (�2/2me)(2π/a)2. The triplets hkl next to each branch refer to the corresponding reciprocal-lattice vector (2π/a)(h, k, l) The figure also shows the band structure calculated in the empty-lattice approximation, along the four high-symmetry directions of the Brillouin zone, 202 20 Electronic Structure of Semiconductors as usual. The lines Δ, Λ, and Σ join the center Γ = (0, 0, 0) with the cen- ter X = (2π/a)(0, 0, 1) of a square face, the center L = (2π/a)(12 , 1 2 , 1 2 ) of a hexagonal face, and an edge center K = (2π/a)(34 , 3 4 , 0) of a hexagonal face, respectively, while another line joins point X ′ = (2π/a)(0, 1, 0) (which is equivalent to X) and point U = (2π/a)(14 , 1, 1 4 ) (which is equivalent to K). The periodic potential modifies this band structure. As far as the behavior of semiconductors is concerned, the most important is to understand what happens at and close to the zone center Γ where an eightfold degenerate state is found above the lowest nondegenerate level. According to the discussion in Chapter 18, the wavefunctions of these eight states can be written as ψnk(r) = 1√ V ei(k+Gi)·r (20.2.1) in the empty-lattice approximation, where Gi = (2π/a) (±1,±1,±1). To de- termine the extent to which this eightfold degeneracy is removed in k = 0, the method described in Chapter 18 for the lifting of accidental degeneracies is applied to the diamond lattice. The little group of point Γ – i.e., the group of those symmetry operations that take Γ into itself or an equivalent point – is the 48-element group Oh. Using the character table of irreducible representations given in Appendix D of Volume 1, the eight-dimensional representation over the eight functions above can be reduced to two one-dimensional (Γ1 and Γ ′2) and two three-dimensional irreducible representations (Γ15 and Γ ′25): Γ = Γ1 + Γ ′ 2 + Γ15 + Γ ′ 25 . (20.2.2) It is also straightforward to find the wavefunctions that transform according to these irreducible representations as linear combinations of the functions eiGi·r. The representation Γ1 is associated with the symmetric combination ψΓ1(r) = 1 8 [ e2πi(x+y+z)/a + e2πi(x+y−z)/a + e2πi(x−y+z)/a + e2πi(−x+y+z)/a+ + e−2πi(x+y+z)/a + e−2πi(x+y−z)/a + e−2πi(x−y+z)/a + e−2πi(−x+y+z)/a ] = cos(2πx/a) cos(2πy/a) cos(2πz/a) . (20.2.3) The combination associated with Γ ′2 is ψΓ ′2(r) = sin(2πx/a) sin(2πy/a) sin(2πz/a) , (20.2.4) while the combinations associated with the three-dimensional representations are ψ (1) Γ15 (r) = sin(2πx/a) cos(2πy/a) cos(2πz/a) , ψ (2) Γ15 (r) = cos(2πx/a) sin(2πy/a) cos(2πz/a) , ψ (3) Γ15 (r) = cos(2πx/a) cos(2πy/a) sin(2πz/a) , (20.2.5) 20.2 Band Structure of Pure Semiconductors 203 and ψ (1) Γ ′25 (r) = cos(2πx/a) sin(2πy/a) sin(2πz/a) , ψ (2) Γ ′25 (r) = sin(2πx/a) cos(2πy/a) sin(2πz/a) , ψ (3) Γ ′25 (r) = sin(2πx/a) sin(2πy/a) cos(2πz/a) . (20.2.6) Hence the eightfold degeneracy is lifted in such a way that two triply de- generate states, of symmetry Γ15 and Γ ′25, and two nondegenerate states, of symmetry Γ1 and Γ ′2, arise. Apart from exceptional cases, their energies are different. We can also examine what happens to the electron states along the lines Δ and Λ close to Γ . To this end we have to make use of the compatibility relations between the irreducible representations that belong to point Γ and lines Δ and Λ, which can be directly established from the character tables. Table 20.6 contains these relations for the relevant representations. Table 20.6. Compatibility relations between irreducible representations for